Variable metric proximal point algorithm for monotone operators

SIAM Journal on Control and Optimization

Volumn 37, Issue 2, 1998, Pages 353-375

  Burke, J V ^a   Qian, Maijian ^b  

^a UNIVERSITY OF WASHINGTON   (United States)

^b CALIFORNIA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; MATHEMATICAL OPERATORS; PROBLEM SOLVING; ROBUSTNESS (CONTROL SYSTEMS);

GLOBAL CONVERGENCE; MAXIMAL MONOTONE OPERATOR; PROXIMAL POINT ALGORITHM;

ALGORITHMS;

EID: 0032202136     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/s0363012992235547     Document Type: Article

References (44)

1. A.D. ALEXANDROV, The existence almost everywhere of the second differential of a convex function and some associated properties of convex surfaces, Ucenye Zapiski Leningr. Gos. Univ. Ser. Mat., 37 (1939), pp. 3-35 (in Russian).
2. J.P. AUBIN AND H. FRANKOWSKA, Set-Valued Analysis, Birkhäuser, Boston, 1990.
3. J.F. BONNANS, J.C. GILBERT, C. LEMARÉCHAL. AND C. SAGASTIZÁBAL, A family of variable metric proximal point methods, Math. Programming, 68 (1995), pp. 15-47.
4. L.M. BREGMAN, The method of successive projection for finding a common point of convex sets, Soviet Mathematics Doklady, 162 (1965), pp. 487-490.
5. J.V. BURKE AND M. QIAN, On the local super-linear convergence of a matrix secant implementation of the variable metric proximal point algorithm for monotone operators, in Reformulation - Nonsmooth, Piecewise Smooth, Semi-smooth, and Smoothing Methods, L. Qi and M. Fukushima, eds., Kluwer Academic Publishers, Norwell, MA, 1998, pp. 317-334.
6. J.V. BURKE AND M. QIAN, On the super-linear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating, Math. Programming, 1999, to appear.
7. X. CHEN AND M. FUKUSHIMA, Proximal Quasi-Newton Methods for Nondifferentiable Convex Optimization, Math. Programming. 1999, to appear.
8. K. DEIMLING, Nonlinear Functional Analysis, Springer-Verlag, New York, 1980.
9. J. ECKSTEIN, Splitting Methods for Monotone Operators with Application to Parallel Optimization, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 1989.
10. M. FUKUSHIMA AND L. QI, A globally and superlinearly convergent algorithm for nonsmooth convex minimization, SIAM J. Optim., 6 (1996), pp. 1106-1120.
11. O. GÜLER, New proximal point algorithms for convex minimization, SIAM J. Optim., 2 (1992), pp. 649-664.
12. S. HAN, A decomposition, method and its application to convex programming, Math. Oper. Res., 14 (1989), pp. 237-248.
13. A. IUSEM, private communication, IMPA, Rio de Janeiro, Brazil, 1996.
14. J.E. DENNIS, JR., AND J.J. MORÉ, A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Comp., 28 (1974), pp. 549-560.
15. J.E. DENNIS, JR., AND R.B. SCHNABEL, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, NJ, 1983.
16. G. KASSAY, The proximal points algorithm for reflexive Banach spaces, Studia Univ. Babes-Bolyai Math., 30 (1930), pp. 9-17.
17. C. LEMARÉCHAL, Bundle methods in nonsmooth optimization, in Nonsmooth Optimization, C. Lemaréchal and R. Mifflin, eds., Pergamon Press, Oxford, 1978.
18. C. LEMARÉCHAL AND C. SAGASTIZÁBAL, An approach to variable metric bundle methods, in IFIP Proceedings, Systems Modeling and Optimization, J. Henry and J.P. Yuan, eds., Springer, Berlin, 1994, pp. 144-162.
19. C. LEMARÉCHAL AND C. SAGASTIZÁBAL, Practical aspects of the Moreau-Yosida regularization: Theoretical preliminaries, SIAM J. Optim., 7 (1997), pp. 367-385.
20. C. LEMARÉCHAL AND C. SAGASTIZÁBAL, Variable metric bundle methods: From conceptual to implementable forms, Math. Programming, 76 (1997), pp. 393-410.
21. F.J. LUQUE, Asymptotic convergence analysis of the proximal point algorithm, SIAM J. Control Optim., 22 (1984), pp. 277-293.
22. B. MARTINET, Regularisation d'inequations variationelles par approximations successive, Revue Française d'Informatique et de Recherche Opérationelle, 4 (1970), pp. 154-158.
23. B. MARTINET, Determination approachée d'un point fixe d'une application pseudo-contractante, cas de l'application prox, Comptes Rendus de l'Académie des Sciences, Paris, Série A, 274 (1972), pp. 163-165.
24. R. MIFFLIN, A quasi-second-order proximal bundle algorithm, Math. Programming, 73 (1996), pp. 51-72.
25. R. MIFFLIN, D. SUN, AND L. QI, Quasi-Newton Bundle-Type Methods for Nondifferentiable Convex Optimization, Technical report AMR 96/21, Dept. of Applied Math., University of New South Wales, Sydney, New South Wales, Australia, 1996.
26. F. MIGNOT, Control dan les inequations variationelles elliptiques, J. Funct. Anal., 22 (1976), pp. 130-185.
27. G.J. MINTY, Monotone (nonlinear) operators in Hubert space, Duke Math. J., 29 (1962), pp. 341-346.
28. J.J. MOREAU, Proximité et dualité dans un espace Hilbertien, Bull. Soc. Math. France, 93 (1965), pp. 273-299.
29. J.M. ORTEGA AND W.G. RHEINBOLDT, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
30. J.-S. PANG AND L. QI, Nonsmooth equations: Motivation and algorithms, SIAM J. Optim., 3 (1993), pp. 443-465.
31. 1 problems, J. Optim. Theory AppL, 85 (1995), pp. 633-648.
32. G.B. PASSTY, Weak convergence theorems for nonexpansive mappings in Bonach spaces, J. Math. Anal. Appl., 67 (1979), pp. 274-276.
33. R.R. PHELPS, Convex Functions, Monotone Operators, and Differentiability, Lecture Notes in Math., Springer-Verlag, New York, 1989.
34. L. QI, Convergence analysis of some algorithms for solving nonsmooth equations, Math. Oper. Res., 18 (1993), pp. 227-244.
35. L. QI, Second-order analysis of the Moreau-Yosida regularization of a convex function, Technical report AMR 94/20, Dept. of Applied Math., University of New South Wales, Sydney, New South Wales, Australia, 1994.
36. L. QI AND X. CHEN, A preconditioning proximal Newton method for nondifferentiable convex optimization, Math. Programming, 76 (1995), pp. 411-430.
37. L. QI AND J. SUN, A nonsmooth version of Newton's method, Math. Programming, 66 (1994), pp. 25-43.
38. M. QIAN, The Variable Metric Proximal Point Algorithm: Theory and Application, Ph.D. thesis, University of Washington, Seattle, WA, 1992.
39. R.T. ROCKAFELLAR, Conjugate Duality and Optimization, SIAM, Philadelphia, 1974.
40. R.T. ROCKAFELLAR, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., 1 (1976), pp. 97-116.
41. R.T. ROCKAFELLAR, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877-898.
42. R.T. ROCKAFELLAR, Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. H. Poincaré Analyse Non Linéaire, 2 (1985), pp. 167-184.
43. J.E. SPINGARN, Partial inverse of a monotone operator, Appl. Math. Optim., 10 (1983), pp. 247-265.
44. J.E. SPINGARN, Applications of the methods of partial inverses to convex programming: Decomposition, Math. Programming, 32 (1985), pp. 199-223.